Unlike the thicker dorsal shell of a trilobite, many of the ventral
(underside) features, including limbs and antennae, usually are not preserved. The ventral portions that are typically preserved include the doublure (a ventral extension of the dorsal exoskeleton),
a special part of the doublure, typically separated by sutures at the anterior
of the cephalon, called the rostral plate,
and a hard mouthpart called the hypostome,
that typically underlies the glabella.

The figure below shows the underside of a typical trilobite fossil,
with cephalic, thoracic, and pygidial doublure,
the rostral plate and associated sutures, and hypostome.
The dark gray area represents the hollow interior of the dorsal shell. In
this case, the hypostome is separated from the rostral plate, which is called
the natant condition. In other species,
the hypostome may be connected to the rostral plate (the conterminant condition), separated only by a suture (as in the ventral reconstruction of Olenoides
serratus below),
or even fused to the rostral plate. These hypostome types are
important in trilobite classification.

If you want more detailed definitions of the terms above, I have
provided a glossary.

In this depiction, the limbs and other ventral parts of the Burgess
shale trilobite Olenoides serratus are shown. As mentioned above,
only very rarely are these structures preserved and fossilized (other than
the hypostome). Long, many-segmented antennae
emerge from lateral notches of the hypostome
(which overlies the mouth), and many pairs of legs,
varying very little except in size, proceed from the cephalon to the
pygidium (three limb pairs under the cephalon, and one pair for each
axial segment in the thorax and pygidium). This primitive lack of
specialization is one of the features of trilobite limbs, shared with
many other Paleozoic Arachnomorpha.

The limbs are attached to a sequential set of axial sternites (ventral segments) bearing a thin, uncalcified exoskeleton. Each of the bases of the limbs
possess jagged toothlike structures that are thought to have processed food
passed between the legs forward to the hypostome and mouth. Such food-processing
limb bases are referred to as gnathobases (gnathos = jaw). (see Trilobite Feeding Behavior.

Between the endopods (crawling limbs) and the body are pairs of
finely branched feathery structures (typically interpreted as gills), borne on the exopods and here colored red, (see additional detail on trilobite
limbs, below). In some trilobites, it is thought that movements of
the exopods might have allowed the animal to swim (as similar movements provide
swimming locomotion in modern marine and freshwater arthropods). Finally,
at the rear of the trilobite, two antenna-like cerci
(sense organs) are depicted, although Olenoides serratus is the only
species among all trilobites that are known to have borne them.

How do we know about the
limbs
and antennae of trilobites?

Very rarely, conditions for preservation are so good that these
delicate features are preserved. Only about twenty different species of trilobites have been found with preservation of antennae and limbs. To the right
is an example of the olenid trilobite Triarthrus eatoni showing preserved
antennae, legs, and gill filaments. The image can be found at Per
Hansson's Trilobite Gallery, which I encourage you to visit!

Similarly, some of the Burgess Shale trilobites, notably Olenoides
serratus, show antennae, limbs, and anal cerci, as in the double specimen
below (image via the Smithsonian Institution). Images such as these were used to create the reconstruction of Olenoides
above.

The limb details of Olenoides show that it was probably
a predator or scavenger, bearing spines and a heavy gnathobase with which
to tear at the tissues of its prey.

The limb details of Triarthrus (below) are much more delicate,
compared to Olenoides ((left), but there are many similarities in
limb structure, notably the number of segments in the endopod (walking leg),
and the delicate gill filaments of the exopod. These similarities were once
considered an important shared character among trilobites and their relatives,
but we now know that many of the Paleozoic arthropods have similar limb
structure, and that limb similarity points to a shared primitive condition.

On the other hand, some trilobites have limbs that seem quite
dissimilar to the pattern above. The limbs of Ceraurus bear an unusual
paddle-like exopod with several distinct segments, while the endopod is remarkably
simple and unadorned.

Perhaps the most unusual of known trilobite limbs are those borne
by Agnostida. They are so unusual that
some scientists cite Agnostine limbs as evidence that they must not be
true trilobites!

A
2003 paper by Nigel Hughes reviews all of the trilobite species with
descriptions of ventral features, presenting a summary table including
the data below. It indicates that the typical trilobite bore a pair of
antennae, then 3 cephalic limbs, followed by trunk (thorax + pygidium)
limbs of variable number, depending on the number of thoracic and
pygidial segments. The typical limb consisted of 6 or 7 podomeres. Olenoides serratus remains the only trilobite with antenniform posterior cerci preserved.

Here is an example specimen of Rhenops cf. anserinus from Hunsruck, Germany, that has been prepared to exposed both dorsal and ventral surfaces.
Limbs and antennae are preserved in pyrite.
Image courtesy of Andreas Ruckert:

At the 2008 Trilobite Conference in Spain, this image of the New York trilobite Triarthrus eatoni won
wide acclaim as the best preserved cephalic ventral features of a
trilobite yet found. The arrangement of the cephalic limbs converging
on the hypostome indicate how the gnathobases act as mouthparts
processing and manipulating food in the vicinity of the mouth, which
underlies the hypostome. Image courtesy of the collection of Dr. Ed
Staver:

Sources cited:

Bartels, C., D.E.G. Briggs, and G. Brassel. 1998. The Fossils of the Hunsruck slate: Marine life in the Devonian. Cambridge Paleontological Series Number 3. Cambridge University Press, Cambridge.